71 International Journal for Modern Trends in Science and Technology
International Journal for Modern Trends in Science and Technology
Volume: 02, Issue No: 11, November 2016http://www.ijmtst.com ISSN: 2455-3778
Comparative Analysis of PID, SMC, SMC with PID Controller for Speed Control of DC Motor
M. Sai Chetana1 | K. Ravi Kumar2 | Ch. Vishnu Chakravarthi3
1PG Student, Department of Electrical and Electronics Engineering, Sanketika Institute of Technology and Management, Visakhapatnam, Andhra Pradesh, India
2Assistant Professor, Department of Electrical and Electronics Engineering, Sanketika Institute of Technology and Management, Visakhapatnam, Andhra Pradesh, India
3Head, Department of Electrical and Electronics Engineering, Sanketika Institute of Technology and Management, Visakhapatnam, Andhra Pradesh, India
To Cite this Article
M. Sai Chetana, K. Ravi Kumar, Ch. Vishnu Chakravarthi, “Comparative Analysis of PID, SMC, SMC with PID Controller for Speed Control of DC Motor ”, International Journal for Modern Trends in Science and Technology, Vol. 02, Issue 11, 2016, pp. 71-76.
In this thesis, sliding mode control (SMC) technique is used to control the speed of DC motor. The performance of the SMC is judged via MATLAB simulations using linear model of the DC motor and known disturbance. SMC is then compared with PID controller. The simulation result shows that the sliding mode controller (SMC) is superior controller than PID for the speed control of DC motor. Since the SMC is robust in presence of disturbances, the desired speed is perfectly tracked. The sliding mode control (SMC)can adapt itself to the parameter variations and external disturbances, problem of chattering parameter, resulting from discontinuous controller, is handled by sliding with smooth control action
KEYWORDS:DC motor, PID controller, Sliding mode controller (SMC)
Copyright © 2016 International Journal for Modern Trends in Science and Technology All rights reserved.
I. INTRODUCTION
DC motors have been widely used in many industrial applications such as electric vehicles, [1] steel rolling mills, electric cranes, robotic manipulators, and home appliances due to precise, wide, simple, and continuous control characteristics. The purpose of a speed controller is to drive the motor at desired speed.
DC motors are generally controlled by conventional [2-12] Proportional plus Integral controllers, since they can be designed easily.
However the performance of PID controller for speed control [10] degrades under external disturbances and machine parameter variations. This makes the use of PID controller
a poor choice for variable speed drive applications.
In the past three decades, nonlinear and adaptive control methods have been used extensively to control DC drives [2]. In these methods, the state estimation and parameter identification are based on and limited to linear models [4]. Performance comparison of sliding mode control and conventional pi controller for speed control of separately excited DC motors.
Here SMC, PID with SMC controller are designed for the DC motor system and their performance is compared [4]
ABSTRACT
72 International Journal for Modern Trends in Science and Technology
II. MODELING OF DCMOTOR
A separately excited dc motor has the simplest decoupled electromagnetic structure [4].A schematic diagram of the separately excited DC motor is shown in follows as description of the system along with the Mathematical model.
Figure 1: A Separately excited DC motor 𝐿𝑎𝑑𝑖𝑎
𝑑𝑡 +𝐼𝑎𝑅𝑎+𝐸𝑏=𝐸𝑎 (1) Where Ea is the Applied Voltage, Ra is the armature resistance, La is the Equivalent armature inductance, Ia current flowing through armature circuit, 𝐸𝑏 is the back emf and, the dynamics of the mechanical system is given by the torque balance equation
J𝑑2θ
𝑑 𝑡2 + B𝑑θ
𝑑𝑡 +𝑇𝑙 =𝑇𝑚 =𝐾𝑡𝐼𝑎 (2) Terminal voltage Va is taken to be the controlling variable. One can write state model with the ω and Ia as state variables and Va as manipulating variable, as given below
𝑒𝑏(t)=𝑘𝑏𝑤(𝑡) (3)
Where Kb is the back emf constant in Vs/rad. The input terminal voltage Va is taken to be the controlling variable. One can write state model with the ω and Ia as state variables and 𝑉𝑎 as manipulating variable, as given below
x1 = θ
x2 = x1 = θ =w x3 =Ia
𝑥 = 𝑤 𝐼𝑎 =
−𝐵 𝐽
𝐾𝑡 𝐽
−𝐾𝑏 𝐽
−𝑅𝑎 𝐿𝑎
𝑥2
𝑥3 + 0
1 𝐿𝑎
𝑉𝑎 (t) (4) PARAMETERS OF THE DC MOTOR
W (s) Va(s) =
K m J
S2+ RL+bJ S+ Rb +K e K m JL
(5) (5) in time domain is as follows
Using the parameters given in Table 1, transfer function of the DC motor with angular velocity as controlled variable and input terminal voltage as manipulating variable is determined as given below
𝑊(𝑠)
𝑉𝑎(𝑠) = 88.76
𝑆2+24𝑆+53.25 (6)
𝑑2𝑤 𝑑𝑡2 + 𝑅
𝐿+𝑏
𝐽 𝑑𝑤
𝑑𝑡+ 𝑅𝑏 +𝐾𝑒𝐾𝑚
𝐽𝐿 𝑤 =𝑘𝑚
𝐽𝐿 w (7)
However, if the state variables consider 𝑥1 =w and 𝑥2= x1 =w .The system described by
equation (4) by equation (8) will be expressed, Where the only variable is the angular velocity and derivative
𝑥 = 0 1 A1 A2 x1
x2 + 0
Km JL
(u) (8) A1 = - Rb +KeKm
JL (9) A2 = - R
L+b
J (10) III. PIDCONTROLLER
Proportional, integral and derivative are the basic modes of PID controller. Proportional mode provides a rapid adjustment of the manipulating variable reduces error and speeds up dynamic response Integral mode achieves zero offset. Derivative mode provides rapid correction based on the rate of change of controlled variable. The controller transfer function is given by 𝐶𝑃𝐼𝐷 (S) = 𝐾𝑃(1+ 1
𝑇𝑠(𝑠)+ 𝑇𝑑s) where, Kp, Ts and Td are the proportional, integral and derivative constants of PID controller respectively. PID controller tuning algorithm is based on Ziegler-Nichols open loop method. And the preference is given to the load disturbance rejection
IV. SLIDING MODE CONTROLLER DESIGN
A linear system can be described in the state space as follows
𝑥 =Ax + Bu (12) Where X ∈ 𝑅𝑛 , U ∈𝑅, A ∈𝑅𝑛∗𝑛,B∈𝑅𝑛 is a full rank matrix Where A and B are controllable matrixes and the functions of state variables are known as switching function
PARAMTERS SPECIFICATIONS VALUE
𝑅𝑎 Armature resistance 1.2 Ώ 𝐿𝑎 Inductance of Armature
winding 0.05H
J Moment of inertia 0.135Kg𝑚2 /𝑠2 B Frictional coefficient 0 Nms 𝐾𝑡 Torque constant 0.06 Nm/A 𝐾𝑏 Back emf constant 0.6 V
73 International Journal for Modern Trends in Science and Technology σ =Sx (13)
The main idea in sliding mode control is
• Designing the switching function so that manifold (sliding mode) provide the desired dynamic ( 𝜎 = 0)
• Finding a controller ensuring sliding mode of the system occurs in finite time First of all, the system should be converted to its regular form 𝑥 =Tx (14) Where T is the matrix that brings the system to it regular form
𝑥1 =𝐴 𝑥11 + 𝐴1 𝑋12 (15) 2 𝑥2 =𝐴 𝑥21 + 𝐴1 𝑥22 +𝐵2 u 2
The switching function in regular form is:
σ =𝑠1 𝑥1 +𝑠2 𝑥2 (16) On the sliding mode manifold ( 𝜎 = 0)
𝑥2= -𝑠2−1𝑠1 𝑥1 (17) From (17) & (15)
𝑥1 =(𝐴11 𝑥1− 𝐴12 𝑠2
−1𝑠1 𝑥1) (18) One of matrixes in product: 𝑠2
−1𝑠1 should be chosen arbitrary. Usually (19) is used to ensure that S2 is invertible
𝑠2 = 𝐵2−1 (19) can be calculated by assigning the Eigen value of (18) by pole placement method. Hence, switching function will be obtained as follows:
S = [𝑠1 𝑠2 ]𝑇 (20) The control rule is:
u =𝑢𝑐+ 𝑢𝑑 (21)
Where 𝑢𝑐and 𝑢𝑑 are continuous and discrete parts,respectively and can be calculated as follows:
𝑢𝑐= - 𝐴 𝑥21 1− 𝐴 σ (22) 22 𝑢𝑑= -𝑘𝑠sign (σ) - 𝑘𝑝(σ) (23) Where sign is sign function. , and are constants calculated regarding to lyapunov stability function We are going to set the angular velocity over a certain value r, so switching function is
σ = 𝑠1(𝑥1 - r) +𝑠2𝑥2 (24) If the controller switching function is designed to be placed on the surface σ =0 then Solving equations (24) assume σ =0 w and 𝑤 and are obtained by
w = r (1 -𝑒 𝑠1𝑠2 𝑡) (25)
𝑤 = r (𝑠1
𝑠2) 𝑒 𝑠1𝑠2 𝑡 (26)
As equation (8) it is regular form, so the transformation matrix is equal to the unit matrix Factor 𝑠2 according to equation (19) must be calculated
𝑠2 =𝐽𝐿
𝐾𝑚 (27) Also according to (12-19) 1 s calculated and w Pole placement method using (12-21) .Suppose we have to placed system poles λ in so we have
𝑠1
𝑠2 = -λ (28) σ = 𝐽𝐿
𝐾𝑚( -λ (w –r)+𝑤 ) (29)
A. CONTROLLER DESIGN:
If the equation (8) can be rewritten based on the state variables and 𝑋1=(𝑥1 -r) the following is reached
𝑋1
𝜎 = 𝐴11 𝐴12 𝐴21 𝐴22
𝑋1 𝜎 + 0
1 𝑈𝑛 (30) 𝐴 = - 11 𝑠1
𝑠2 = -λ
𝐴 = 12 1
𝑠2
𝐴 = 𝐴21 1𝑆2− 𝐴2𝑆1 -𝑆1
2
𝑆2 =(𝑆2(𝐴1+𝐴2𝜆 − 𝜆2) 𝐴 = 𝐴22 2+𝑆1
𝑆2
𝑢𝑛=𝑠2−1u +𝐴1r (31) Thus the relations (21), (22) and (23) controller for the system (30) is designed as follows
𝑢𝑛= -𝐴 𝑋21 1− 𝐴 𝜎 -𝑘22 𝑠sigm (σ) - 𝑘𝑝(σ) (32) The below equation Sets armature voltage feedback based on the derivative of the angular velocity for motor.
u = 𝑆2[𝐴1𝑟 + 𝑆2(𝐴1+𝐴2𝜆 − 𝜆2)(w –r) + ( 𝐴2− 𝜆)σ+
𝑘𝑠sgn (σ) + 𝑘𝑝(σ) (33) So the sliding mode controller is
u = 𝐽𝐿
𝐾𝑚
𝑅𝑏 +𝐾𝑒𝐾𝑚 𝐽𝐿 w +𝐽𝐿
𝐾𝑚
𝑅𝑏 +𝐾𝑒𝐾𝑚 𝐽𝐿 + λ 𝑅
𝐿+𝑏
𝐽 +𝜆2 𝑅
𝐿+𝑏
𝐽 + 𝑅𝑏 +𝐾𝑒𝐾𝑚
𝐽𝐿 (w-r) + 𝑅
𝐿+𝑏
𝐽 λ-𝑘𝑠sign(σ) - 𝑘𝑝(σ) (34)
74 International Journal for Modern Trends in Science and Technology Switching function of sliding mode controller for
DC motor control method according to the relations (34) and (33) are designed .If the motor parameters like table (1),then the controller we will numerically designed as follows
σ = -1.12(w –r)+0.0112𝑤
(35)
After solving The controller u is given by
U =0.0112(53.24)w + 0.0112(53.25)(w-r) -124(σ) –sgn(s) (36)
Where λ, ks and kp parameters are -100, 1 and 0 respectively
V. RESULTS AND OUTPUTS
The DC motor, a PID controller is attached and the corresponding simulink model and its output for the same reference input of 1000rpm is given below
Figure 1: Simulink model of DC motor
Speed Vs Time
Figure 2: Speed response of DC motor The DC motor, attached and the corresponding simulink model and its output for the same reference input of 1000rpm is given below
Figure 1: Simulink model of DC motor with PID
The DC motor, a PID controller is attached and the corresponding simulink model and its output for the same reference input of 1000rpm is given below
Speed Vs Time
Figure 3: Speed response of DC motor with PID Controller
The DC motor, a SMC controller is attached and the corresponding simulink model and its output for the same reference input of 1000rpm is given below
Figure 4: simulink model of dc motor with SMC controller
Speed Vs Time
Figure 5 :Speed response of DC motor is done by SMC
Comparing the PID,SMC,SMC with PID is attached to DC motor with a reference speed of 1000 rpm The outputs are compared based on the settling time
75 International Journal for Modern Trends in Science and Technology Figure 6: Simulink model of DC motor of
combined with PID, SMC, SMC with PID
Speed Vs Time
Figure 7: Speed response of DC motor is done by using PID controller and SMC, the response due to SMC is better compared to PID controller SMC does not vary with parameter variations, by varying the parameters of R, L, J of DC motor, by increasing the percentage of R, L, J parameters of DC motor with speed of 1000 rpm
Figure 8: Comparison of internal parameters of DC motor
Speed Vs Time
Figure 9:Speed response of DC motor is observed by varying the internal parameters R, L, J of the DC motor
Figure 10: Simulink model of DC motor is observed when external disturbance is added
Figure 11: Speed response of DC motor is observed when external disturbance is added
COMPARISION TABLE Controller 𝐓𝐬
(sec) OVERSHOOT DISTURABNCE REJECTION
PID 0.3 PRESENT POOR
SMC 0.2 NIL GOOD
SMC WITH
PID 0.07 NIL GOOD
Where Ts = settling time
VI. CONCLUSION
In this paper sliding mode control (SMC) Proposed to speed control of DC motor. At first for controlling speed of DC motor a simplified closed loop is utilized. Then DC motor is modeled after that speed controller is designed. As sliding mode control is based on the system Dynamic characteristics also it took a lack of influence of external disturbances from user as result it worked more useful and results confirms that used sliding mode control for speed control is more efficient in comparison with PID controller.
ACKNOWLEDGMENT
The authors would like to express their gratitude to Dr.S.V.H Rajendra, Secretary, AlwarDasGroup of Educational Institutions, SriVBhaskar, Dean for
76 International Journal for Modern Trends in Science and Technology their encouragement and support throughout the
course of work. The authors are grateful to Dr.N.C.Anil, Principal, Sanketika Institute of Technology and Management, EEE Department and staff for providing the facilities for publication of the paper.
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